Welfare Maximizing Contest Success Functions when the Planner Cannot Commit

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Munich Personal RePEc Archive

Welfare Maximizing Contest Success Functions when the Planner Cannot Commit

Corchon, Luis and Dahm, Matthias

Carlos III

12 November 2009

Online at https://mpra.ub.uni-muenchen.de/20074/

MPRA Paper No. 20074, posted 18 Jan 2010 10:51 UTC

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Working Paper 09-73 Departamento de Economía

Economic Series (43) Universidad Carlos III de Madrid

November 2009 Calle Madrid, 126

28903 Getafe (Spain) Fax (34) 916249875

Welfare Maximizing Contest Success Functions when the Planner Cannot Commit

^∗

Luis Corchón^Υand Matthias Dahm^Ζ November 12, 2009

Abstract

We analyze how a contest organizer chooses the winner when the contestants.efforts are already exerted and commitment to the use of a given contest success function is not possible. We define the notion of rationalizability in mixed-strategies to capture such a situation. Our approach allows to derive different contest success functions depending on the aims and attitudes of the decider. We derive contest success functions which are closely related to commonly used functions providing new support for them. By taking into account social welfare considerations our approach bridges the contest literature and the recent literature on political economy.

Keywords: Endogenous Contests, Contest Success Function, Mixed-Strategies.

JEL Classification: C72 (Noncooperative Games), D72 (Economic Models of Political Processes: Rent-Seeking, Elections), D74 (Conflict; Conflict Resolution; Alliances).

∗ We wish to thank Carmen Beviá, Amihai Glazer, Carolina Manzano, Stergios Skaperdas, Bernd Theilen and Galina Zudenkova

for helpful discussions and suggestions. The usual disclaimer applies. This work was carried out while Dahm was visiting the University of California, Irvine. The hospitality of this institution and the financial support through the Spanish program José Castillejo is gratefully acknowledged. The first author acknowledges financial support from the Spanish Ministerio de Educación y Ciencia, project SEJ2005-06167/ECON and the second acknowledges the support of the Barcelona GSE, the Government of Catalonia and the Spanish Ministerio de Educación y Ciencia, project SEJ2007-67580-C02-01.

Υ Departamento de Economía. Universidad Carlos III de Madrid. Calle Madrid, 126. 28903 Getafe (Madrid). Spain. E-mail:

lcorchon@eco.uc3m.es. Phone: +34 916 249617. Fax: +34 916 249875.

Ζ Departamento de Economía. Universitat Rovira i Virgili. Avenida de la Universitat, 1. 43204 Reus (Tarragona). Spain. E-mail:

matthias.dahm@urv.cat. Phone: +34 977 758 903. Fax: +34 977 759 810.

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1 Introduction

In a contest players exert e¤ort in order to win a certain prize. Contests have been used to analyze a variety of strategic situations ranging from rent-seeking and lobbying to con‡ict, arms races, warfare, and promotional e¤orts, as well as to sports.¹

In many of these situations there is a contest organizer who awards the prize.² For example in litigation a court decides on the winner; in lobbying, rent-seeking and rent-defending contests bureaucrats or politicians award a prize; in internal labor market tournaments jobs are allocated by the manager of an organization; or in beauty contests the decision where to locate an event is taken by a committee.

A crucial determinant for the equilibrium predictions of contests is the speci…cation of the so-called contest success function (CSF) which relates the players’ e¤orts and win probabilities (also interpreted as shares of the prize).³ In this paper we provide a framework in which CSFs are derived as an optimal choice of the contest administrator.⁴ See our remarks in the

…nal section regarding the interpretation of our results when the contest is not organized by a contest administrator. The administrator is unable to commit to a CSF so she chooses the probabilities with which the prize is given in order to maximize her utility given the choices of the contestants. Assuming complete information, contestants anticipate the choice made by the contest organizer. We apply this framework by postulating di¤erent utility functions for the decider. Of course the plausibility of our results hinges on the plausibility of these utility functions. Thus we decide to use only the most popular utility functions in the theory under risk: expected utility and (a special case of) prospect theory. We also follow the recent political economy literature (see Grossman and Helpman, 2001; Persson and Tabellini, 2000) and let the decider care about social welfare - albeit in contrast to that literature in the form of a generalized utilitarian planner. Somewhat surprisingly, this allows us to derive the three most commonly used types of CSFs: non-deterministic CSFs with an additive structure (Proposition 5), deterministic CSFs (Proposition 4 and Example 1) and di¤erence-form CSFs (Propositions 1, 2 and 3).

Our approach di¤ers from other models in which commitment is possible such as the menu auction approach (see Grossman and Helpman, 1994). Dasgupta and Nti (1998) analyzed the optimal design of a contest when the contestants have identical valuations, the planner might retain the prize and when the contest success function must be of the class axiomatized by Skaperdas (1996) (see equation (2) below). They …nd that, when the planner has a low valuation

1For a survey see Konrad (2007).

2We use the terms contest administrator, contest organizer, decider and planner interchangeably. Participants in the contest are called contestants, rent seekers or contenders.

3A prominent example of this interpretation as shares is Wärneryd (1998). He analyzes a contest among jurisdictions for shares of the GNP and compares di¤erent types of jurisdictional organization.

4Other alternative foundations for CSFs are provided by Blavatskyy (2008), Münster (2009) and Rai and Sarin (2009) who o¤er axiomatic characterizations of contest success functions following the seminal paper by Skaperdas (1996). The earlier work on foundations for contest success functions has been extensively reviewed by Konrad (2007).

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for the prize, the optimal CSF is such that the probability that a contestant obtains the prize is the ratio of her e¤ort with respect to total e¤ort. A closely related paper is Epstein and Nitzan (2006) which with respect to commitment can be seen as the benchmark that is the opposite of our paper. In Epstein and Nitzan the contest organizer decides …rst whether to have a contest at all, and if a contest takes place, she chooses the CSF among the elements of a …xed set of CSFs maximizing ex-ante utility. In doing so, the organizer anticipates the equilibrium e¤orts of contestants and needs to be able to commit to employing a given CSF once e¤orts are exerted.

In contrast, in our approach the organizer decides on the winner once e¤orts are exerted.⁵ There are further approaches motivating or endogenizing the CSF and some of them imply that it is not necessary to assume commitment. In one recent approach contestants might be uncertain about a characteristic of the organizer and as a result view the determination of the winner as probabilistic although the organizer chooses in a deterministic way (Corchón and Dahm, 2009;

Skaperdas and Vaidya, 2009). In another, e¤orts are a¤ected by exogenous shocks so that the performance of contestants is di¤erent from e¤orts which generates randomness from the point of view of contestants (Jia 2007).

Our approach cannot be construed as criticism of the commitment assumption. The commitment case is an important benchmark case. It is, however, not always clear how the decider can be trusted to maintain her word. In other cases the planner may prefer a policy of wait and see instead of announcing a certain CSF. Therefore, it is important to know what happens when, after contestants have exerted e¤orts, the contest administrator is no longer constrained by her word and could choose the winner in a di¤erent way. Thus our paper can be viewed as a check on the properties of CSFs in case commitment becomes unlikely. Our main conclusion is that well-known CSFs arise under natural speci…cations of the preferences of the contest organizer.

Thus our approach could be considered as a back up for the use of well-known CSFs.

2 Preliminaries

A contest administrator conducts a contest amongncontestants denoted byi2N :=f1; :::; ng.

Each contestant has a valuation for the prize, denoted by V_i 2R+, and exerts e¤ort G_i 2R+

in order to a¤ect the probability of winning the prize which is given by the CSF.

Formally, a contest success function p(G) = (p₁(G); p₂(G):::; p_n(G)) associates, to each vector of e¤ortsG, a lottery specifying for each agent a probabilityp_iof getting the prize. That is,p_i =p_i(G) is such that, for each contestanti2N,p_i(G) 0, andP_n

i=1p_i(G) = 1.

We say that a CSF is imperfectly discriminating if G_i > 0 implies that p_i(G) > 0.⁶ An

5There are further di¤erences to our paper. In our approach the contest organizer is completely unconstrained in her choice of the contest success function, rather than choosing among the elements of a …xed set of contest success functions. Also, our approach is not restricted to the case of two contestants.

6This is essentially axiom 1 in Skaperdas (1996). The name of this property refers to the fact that a contest can be interpreted as an auction where the prize is auctioned among the agents and e¤orts are bids. In standard auctions the highest bid obtains the prize with probability one. Here, any positive bid entitles the bidder with a positive probability to obtain the object, so it is as if the bidding mechanism did not discriminate perfectly

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example for such a function is the most commonly used CSF introduced by Tullock (1980) which is given by

pi= G^R_i Pn

j=1G^R_j ; fori= 1; :::; n; (1) whereR is a positive parameter. A generalization of this form is

p_i = f_i(G_i) P_n

j=1f_j(G_j); fori= 1; :::; n; (2) where thefi( )are positive increasing functions of its argument. For the case in whichfi( ) =f( ) for i = 1; :::; n, (2) has been axiomatized by Skaperdas (1996). A di¤erent class of CSFs are di¤erence-forms (Hirshleifer, 1989; Baik, 1998; Che and Gale; 2000). The linear di¤erence-form contest in Che and Gale (2000) is de…ned as

p_i = max min 1

2+s(G_i G_j);1 ;0 fori= 1;2 andj 6=i, (3) where s is a positive scalar. Notice that the linear di¤erence-form is not imperfectly discriminating. Another example for a not imperfectly discriminating CSF is a function that assigns probability one to the contestant exerting the highest e¤ort, like in an all-pay auction.

Contestants are risk-neutral. De…ning a_i(G) = 1 when the contest is all-pay (e¤ort is irreversible so that all contestants pay their bid) anda_i(G) =p_i(G)when the contest is winner- pay (e¤ort is like a promise so that only the winner pays his bid), the expected utility of a contestant is

u_i(p_i;G) =p_i(G)V_i a_i(G)G_i:

While contests are usually analyzed as all-pay, winner-pay contests have been analyzed in Skaper- das and Gan (1995), Wärneryd (2000) and Yates (2007). There is also a large literature on the

…rst-price (sealed bid) auction which constitutes the extreme case of a winner-pay contest in which the highest bidder wins with probability one.

The timing is as follows. In an all-pay contest contenders exert e¤ort in the …rst stage (simultaneously), while the administrator assigns win probabilities or shares of the prize in the second stage. In a winner-pay contest contenders promise e¤ort in the …rst stage (simultaneously) but e¤ort is not exerted yet. In the second stage the organizer determines the outcome of the contest. In the third stage the winner exerts the e¤orts promised.⁷ When contestants play mixed strategies, in stage 1 contestants anticipate the CSF and choose a mixed strategy. Then the actual e¤ort levels are realized and the organizer observes them. In stage two the organizer chooses (given the realization of e¤orts) the CSF which is the one contestants have anticipated in stage 1.

among bids.

7Our approach also works if in the …rst stage contestants exert e¤ort sequentially (in the case of the all-pay contest) or promise e¤ort sequentially (in the case of the winner-pay contest). We assume that in a winner-pay contest the promise of e¤ort in stage one will be ful…lled in stage three. A reason for this could be that the prize will only be delivered when e¤ort has been exerted.

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3 Rationalizability in Mixed-Strategies

In this section we formalize the idea that the contest organizer’s choice in the second stage is optimal from her view point. Suppose that, after e¤ort has been exerted or promised in the

…rst stage, the decider assigns win probabilities or shares of the prize maximizing her objective function. Since there is no uncertainty and contestants know the organizer and her incentives, rent-seekers are able to anticipate in the …rst stage which CSF will be chosen.

We capture the idea that a particular CSF, which is used when contestants make their e¤ort choices, is also optimal from the point of view of the contest organizer with the following de…nition. Denote by Sⁿ then 1dimensional simplex.

De…nition 1 The contest success function p(G) = (p₁(G); p₂(G):::; p_n(G)) is rationalizable in mixed-strategies if there is a function W(p;G) such that for all G2Rⁿ,

p(G) =argmax W(p;G),p2Sⁿ.

Here the term mixed-strategies refers to the fact that technically a CSF is a probability distribution even though, as we have already mentioned, it might be interpreted as shares of the prize. In our approach, the contest success function is the best reply of a contest organizer with payo¤ function W(p;G). As we will see in the sequel, the payo¤ function allows to take into account a variety of objectives and attitudes of the decider. The use of mixed-strategies here might be motivated analogously to the classical argument in favor of mixed-strategies, namely that mixed-strategies produce unpredictable choices that cannot be exploited by an opponent.

This might translate to enabling the organizer to avoid that contestants forecast perfectly her choice, which might reduce incentives to exert e¤ort. There are, however, di¤erences between our approach and the way mixed-strategies are usually employed. Usually a player chooses a mixed- strategy when she is indi¤erent between the pure strategies involved. In contrast, we will be able to derive mixed-strategies which are strictly preferred to any other pure or mixed-strategy.

This is because our planners are not expected utility maximizers.

4 Generalized Utilitarian Planner

As commonly assumed in the recent political economy literature (see Grossman and Helpman, 2001; Persson and Tabellini, 2000) let the decider’s objective function depend on expected social welfare.⁸ Consider a generalized utilitarian planner whose payo¤ function is a constant elasticity of substitution function

W(p;G) = 8

>>

<

>>

:

n

X

i=1

(u_i(p_i;G))¹ ^r

!1=(1 r)

if r6= 1

n

X

i=1

ln (u_i(p_i;G)) if r= 1

. (4)

8In that literature the objective function is a weighted average of social welfare and lobbying e¤orts.

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The positive parameterr represents the degree of inequality aversion of the planner. If the CSF speci…es win probabilities, inequality can be interpreted as referring to expected utility. The utilitarian case corresponds to r = 0. When r = 1, the Bernoulli-Nash case obtains. When r goes to in…nity, the Rawlsian case arises and the concern is with the least well-o¤ only.

We consider an all-pay contest.⁹ The planner maximizes W(p;G) as de…ned in (4) with respect top. It is instructive to start with the Bernoulli-Nash case in whichr = 1.

Proposition 1 Suppose the contest is all-pay and maximizing (4) with respect to p has an interior solution. The contest success function

p_i =

1 Xn

j=1(G_j=V_j)

n +G_i=V_i for i= 1; :::; n (5)

is the unique one that can be rationalized in mixed-strategies by a function ful…lling (4) with r= 1.

Proof. Notice …rst that the objective function is strictly concave, as the Hessian matrix is a diagonal matrix which has as itsiith element (V_i)²=(p_iV_i G_i)². Since the solution is interior, the …rst order conditions imply

(p_iV_i G_i)V_j = (p_jV_j G_j)V_i; fori; j= 1; :::; n.

Rearranging we obtain

p_i = p_nV_n G_n V_n +G_i

V_i fori= 1; :::; n.

Adding up over all contestants yields

pn=

1 Xn

j=1(G_j=V_j)

n +Gn=Vn

and replacing this in the previous equation we obtain (5).

Notice that the expression in (5) is a generalized di¤erence-form. To see this consider the following corollary.

Corollary 1 Suppose the contest is all-pay and maximizing (4) with respect to phas an interior solution.

1. If there is a common value V, the contest success function

p_i= 1 n+ 1

nV 0

@(n 1)G_i X

j6=i

G_j 1

A for i= 1; :::; n (6) is the unique one that can be rationalized in mixed-strategies by a function ful…lling (4) withr = 1.

9We omit the discussion of winner-pay contests as the mathematical structure of the problem is closely related to Proposition 5. See the discussion after that proposition.

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2. If n= 2, the contest success function p_i= 1

2 +1 2

G_i Vi

G_j Vj

for i= 1;2 and j6=i (7) is the unique one that can be rationalized in mixed-strategies by a function ful…lling (4) withr = 1.

Notice that the CSFs in (6) and (7) are very related to the linear di¤erence-form contest in Che and Gale (2000) given in (3). When there are two contestants and whenpi >0fori= 1;2, then (5) coincides with (3) if there is a common value V and V = 1=(2s). This threshold V = 1=(2s) has the interesting interpretation that a contestant can only guarantee success by exerting at least this amount more than the opponent.

As Baik (1998) and Che and Gale (2000) have shown, in many di¤erence-form contests there are problems concerning the existence of pure strategy equilibria. But, as we already discussed, our approach can also work when contestants play a mixed-strategy.

We consider now the case in which r6= 1 and r 2(0;1).

p_i =

1 Xn

j=1(Gj=Vj) Xn

j=1(Vj=Vi)¹^r^r

+G_i=V_i for i= 1; :::; n (8)

is the unique one that can be rationalized in mixed-strategies by a function ful…lling (4) with r6= 1 andr 2(0;1).

Proof. Supposer6= 1 and r2(0;1) and consider W⁰(p;G) =

n

X

i=1

(p_iV_i G_i)¹ ^r:

Notice that for r < 1, W(p;G) is an increasing transformation of W⁰(p;G), while for r > 1, W(p;G) is a decreasing transformation of W⁰(p;G). The Hessian matrix of W⁰(p;G) is a diagonal matrix which has as itsiith element r(1 r)(V_i)²(p_iV_i G_i) ^r ¹. W⁰(p;G)is, hence, strictly concave for r <1and strictly convex for r >1. W⁰(p;G) has thus a unique maximizer in the …rst case and a unique minimizer in the latter case. Both correspond to the unique maximizer ofW(p;G).

Let us maximize W⁰(p;G) as de…ned above subject to p 2 Sⁿ, where Sⁿ is the n 1 dimensional simplex. Since the solution is interior, the …rst order conditions imply

(p_iV_i G_i)^rV_j = (p_jV_j G_j)^rV_i; fori; j= 1; :::; n.

Rearranging we obtain

p_i = p_nV_n G_n

(V_n)^1=r (V_i)¹^r^r +G_i Vi

fori= 1; :::; n.

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pn=

1 Xn

j=1(G_j=V_j) Xn

j=1(V_j)¹^r^r (Vn)¹^r^r +Gn=Vn

and replacing this in the previous equation we obtain (8).

Notice that when r goes to one, (8) becomes (5), while when there is a common value (8) becomes (6). Consider now the Rawlsian case in which r goes to in…nity.

p_i =

1 Xn

j=1(Gj=Vj) Xn

j=1V_i=V_j +G_i=V_i for i= 1; :::; n (9) is the unique one that can be rationalized in mixed-strategies by a function ful…lling (4) with r=1.

Proof. Forr equal to in…nity the objective function (4) becomes

W(p;G) = minfp1V1 G1; :::; pnVn Gng: (10) If the solution is interior, it must hold that

p_iV_i G_i =p_jV_j G_j; fori; j = 1; :::; n.

This implies that

p_i = p_nV_n G_n V_i +G_i

V_i fori= 1; :::; n.

p_n=

1 Xn

j=1(G_j=V_j) Xn

j=11=V_j (V_n) ¹+G_n=V_n and replacing this in the previous equation we obtain (9).

Notice that whenr goes to in…nity, (8) becomes (9). Note also that when there is a common value all cases considered so far yield the same CSF.

Corollary 2 Suppose the contest is all-pay, maximizing (4) with respect to p has an interior solution and there is a common value. The contest success function (6) is the unique one that can be rationalized in mixed-strategies by a function ful…lling (4) with r >0.

We may also consider an individual rationality constraint in the planner’s problem adding the restrictions thatu_i 0; i= 1;2; :::; n. Notice that in the cases considered in Propositions 1,

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2 and 3 as long asXn

j=1Gj=Vj 1it can be easily shown that there is a subset of the simplex from which win probabilities can be chosen such that the individual rationality constraint is satis…ed. Note also that in the case of a common value the assumption that the sum of individual e¤orts weighted by the valuation must be smaller than one says that the rent is not completely dissipated. The assumption that Xn

j=1G_j=V_j 1 together with G_i=V_i > 0; i = 1;2; :::; n is also su¢cient to guarantee that maximizing (4) with respect to phas an interior solution. On the one hand, it is immediate that under these assumptionsp_i >0,i= 1;2; :::; n. On the other hand, computing the sum of the win probabilities using (8) shows thatXn

j=1p_j = 1. Thus, we have that pi 2(0;1); i= 1;2; :::; n. This shows that our interiority assumption can be satis…ed under certain conditions.

Finally let us consider the case of r = 0 in which the planner is utilitarian. In this case, in general, there will be no interior solution and the assignment of win probabilities depends only on valuations. More precisely, the planner prefers that the contestant with the highest valuation wins the prize. In the common value case the objective function of the planner becomes insensitive to whom wins the contest.

Proposition 4 Suppose the contest is all-pay and W(p;G)follows (4) with r= 0. The contest success function is such that the contestant with the highest valuation wins the prize: If p_i >0, thenV_i = maxfV₁; V₂; :::; V_ng.

5 Other Approaches

Suppose that the prize is the right to supply a certain good (i.e. Olympic Games) and that the quality of this good is positively related with the e¤ort made by the winner. Let f_i(G_i) be the quality of the prize if agent i wins the contest. Then, expected quality, identi…ed with the utility of the planner is W(p;G) = Xn

i=ip_if_i(G). Since expected utility theory has a linear structure similar to the utilitarian planner of the previous section, imperfectly discriminating CSFs cannot be rationalized.¹⁰ This is illustrated by the following example.

Example 1 Incumbency advantage (Konrad (2002)). Let the contest be winner-pay andn= 2.

Assume f₁(G₁) =G₁ andf₂(G₂) =bG₂ a, where b2(0;1] anda 0. Whena= 0 andb= 1, this function rationalizes the standard …rst-price (sealed bid) auction in mixed-strategies. For other parameter values it rationalizes a biased version of it.

1 0It is also di¢cult to derive imperfectly discriminating CSFs using expected utility theory when the contest is all-pay. Suppose the organizer obtainsf_i(G) =X_n

j=1G_jno matter which contestant wins, then the organizer’s payo¤s do not depend on the assignment rule. The organizer is, thus, indi¤erent between CSFs and any rule including imperfectly discriminating ones can be rationalized in mixed-strategies. But notice that fi(G) does not need to be the same for all i, in which case imperfectly discriminating CSFs cannot be generated. To see this assume that p(G) is imperfectly discriminating and p(G) can be rationalized in mixed-strategies. Then there existG^ 2Rⁿ++ andi; j2N such thatfi(G)^ > fj(G)^ and maximizingW(p;G)^ requiresfj(G) = 0. This^ contradicts thatp(G)is imperfectly discriminating.

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In order to derive imperfectly discriminating CSFs one needs thus to assume some form of non-expected utility theory.

Consider prospect theory. A prospect is an utility level f_i(G_i), which we identify with the quality of the prize, and a probabilitypi. In prospect theory the former is transformed through a value function, while the latter enters the utility through a weighting function. Both functions are assumed to be power functions.¹¹

More precisely, consider the following functional form for W(p;G) which corresponds to a special case of the class postulated by Kahneman and Tversky (1979, p. 276) for regular prospects, namely¹²

W(p;G) =

n

X

i=1

p_if_i(G_i)¹ ,1> >0. (11) Notice that (11) is the sum of n terms—one associated to each contestant. In each such term both components of a prospect are combined in a Cobb-Douglas way under the assumption of constant returns to scale.

Finally, all these contestants’ speci…c Cobb–Douglas functions are aggregated in an additive way. This re‡ects that contestants are perfect substitutes from the contest organizer’s point of view and implies that the marginal product of a contestant’s e¤ort does not depend on the e¤ort of others. Notice lastly that except for the contestants’ speci…c fi( )’s, (11) is symmetric in contestants. For example, the exponents and 1 , which measure the elasticity of the contest organizer’s payo¤ with respect to e¤ort and win probability, take the same value for every participant. Moreover, there is no contestants’ speci…c scaling parameter. Now we have the following result:

Proposition 5 The contest success function of the form (2) is the unique one that can be rationalized in mixed-strategies by a function ful…lling (11).

Proof. Let us maximize W(p;G), p 2 S. Since W(p;G) is continuous on p and S is compact, a maximum exists. Since W(p;G) is strictly concave on p and S is convex the maximum is unique. Consider the …rst order conditions of the maximization with respect top

p_i ¹f_i(G_i)¹ = 0,i= 1;2; :::; n.

Clearly, the maximum is interior because ifpi !0, the left hand side of the above equation goes to in…nity. The above equations imply that

p_i ¹f_i(G_i)¹ =p_j ¹f_j(G_j)¹ ,i; j= 1;2; :::; n.

1 1Value functions in the form of power functions are often used. For a discussion and axiomatic analysis of the so-called probability weighting functions, including the power function employed in (11), see Prelec (1998). Of course, an important di¤erence between our setting and standard applications of prospect theory is that here win probabilities are an object of choice of the decider.

1 2Although Kahneman and Tversky considern= 2, they consider the extension to more outcomes “straight- forward” (p. 288).

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Which yield

p_i = f_i(G_i)p_n

fn(Gn) ,i= 1;2; :::; n.

Substituting these equations in the simplex we obtain that p_n= f_n(G_n)

n

X

j=1

f_j(G_j) .

Substituting this equation in the previous one we obtain (2).

Notice that besides prospect theory there might be other interpretations of (11) that depending on the context might be meaningful. For instance, by choosing f_i(G_i) = (G_i)¹ and by applying to (11) the increasing transformation V(p;G) = (W(p;G))¹⁼ , we obtain the CES utility function with the form

V(p;G) =

n

X

i=1

(piGi)

!1=

with = 1 1

, (12)

where is the elasticity of substitution which is usually assumed to be strictly larger than one.

Since the maximizer is not a¤ected, we obtain p_i = (G_i) ¹= 0

@

n

X

j=1

(G_j) ¹ 1

A, which allows to

explain the exponents in this CSF as the elasticity of substitution of the decider.¹³ In another example one might proceed similarly and choosef_i(G_i) = (V_i G_i)¹ . This corresponds to (4) under the assumption of a winner pay contest.

6 Concluding Remarks

We have shown that contest success functions can be viewed as optimal choices of a contest organizer, given e¤orts of contestants. This implies that it is not necessary to assume that a contest organizer is able to commit to employ a given contest success function. This approach does not rely on uncertainty or reputation e¤ects. We have shown that our approach works both for all-pay and winner-pay contests. It can also motivate both interpretations of contest success functions: win probabilities and shares of the prize. Our exercise yields contest success functions that were already ‘popular’ providing new support for them.

Notice that the our approach is very much related to at least two classic problems.

1 3Under the assumption that the CSF assigns shares of the prize, the CES function indicates a taste for variety.

A politician might have a taste for variety when the lobbying e¤orts are based on di¤erent expertise e.g. when they are used to draft legislation. One interpretation of equation (12) is then that lobbies o¤er to invest e¤ortGi

in drafting a complete bill and that the decider prefers to have parts of the legislation drafted by di¤erent lobbies.

Notice that in order to capture a taste for variety the sum in equation (12) cannot be linear in shares, because

n

X

i=1

p_i (G)

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The …rst classic problem is the equivalence between revealed preference and utility maximizing choices. For a long time our profession struggled to …nd a condition under which these two approaches were equivalent, a …nal solution being obtained by Kihlstrom, Mas-Colell and Sonnenschein (1976). In our case we show the equivalence between contest success functions whose axiomatic properties were known and those arising from the actions of a benevolent and well-informed planner.

The second classic problem is the equivalence between market equilibrium and welfare optimum, i.e. the so-called two fundamental theorems of welfare economics. These theorems assert the equivalence between market equilibrium and allocations obtained by a benevolent and well- informed planner, see Mas-Colell (1985 and 1986). In our case, we …nd that some contest success functions maximize expected social welfare. As in the case of the two fundamental theorems of welfare economics we do not mean that the planner actually exists. The planner is just a surro- gate of what the system achieves by its own forces. Thus, in our case, contest success functions can be determined by the pure form of con‡ict, random elements, etc. What our results say is what these contest success functions are like if they were chosen by a planner (with a visible hand) in order to maximize social welfare.

An interesting implication of our analysis is that contest success functions might depend on the valuations of contestants. This is reasonable when the decider takes into account the welfare e¤ects of her decision on contestants which in a contest model depend on valuations. In this sense our analysis bridges the gap between the contest literature and the recent literature on political economy in which the decider takes into account the welfare of rent-seekers (see Grossman and Helpman (2001)).

By postulating reasonable aims for the decider we have derived commonly used contest success functions and new ones with a mathematical structure similar to popular ones. Future research might postulate further payo¤ functions for the contest organizer and investigate the consequences for the contest success functions which arise.

In contrast, in the contest literature many studies suppose that the contest organizer is only interested in maximizing total expected e¤ort. But as Konrad (2007, p. 69) writes “this is, at best, an approximation to what contest organizers care about in many applications.” In the present paper we have postulated alternative aims for the contest organizer and shown how these aims translate into contest success functions. We hope that our approach opens the door to taking into account further variables in‡uencing contest organizers.

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